For two functions $$f ( x ) = \frac { 1 } { 3 } x ( 4 - x ) , \quad g ( x ) = | x - 1 | - 1$$ let $S$ denote the area enclosed by their graphs. Find the value of $4 S$. [4 points]

123. The area of the region bounded by the graph of the function $y = x^2|x|$ and the line $y = 8$ is which of the following?
(1) $16$ (2) $18$ (3) $22$ (4) $24$

1-1. What is the area of the region bounded by the graphs of the two functions $y = 5 - |x - 1|$ and $y = |x|$?
(1) $8$ (2) $9$ (3) $15$ (4) $12$

1-2. A completely calm balloon loses 5 percent of its air per day. After several days, half of the initial air remains. How many days does it take? $(\log 19 = 1.287,\ \log 2 = 0.301)$
(1) $17$ (2) $18.5$ (3) $21.5$ (4) $25$

1-3. From the equation $\log(x+2) + \log(2x-1) = \log(4x+1)$, what is the value of $\log(5x+2)$ in base 4?
(1) $0.5$ (2) $0.75$ (3) $1.25$ (4) $1.5$

1-4. The graph below shows the function $y = a + b\cos\!\left(\dfrac{\pi}{2}x\right)$, with period $(4,\ 0)$. What is $b$?

[Figure: Graph of a cosine-based function with maximum value $4$ and minimum value near $0$, symmetric about the y-axis]
(1) $-2$ (2) $-1$ (3) $1$ (4) $2$

1-5. How many distinct real roots does the equation $2 = (x^2 - 2x)^2 - (x^2 - 2x)$ have?
(1) $1$ (2) $2$ (3) $3$ (4) $4$

1-6. If $f(x) = x + |x|$ and $g(x) = |x+1| + 1$, then the range of $\left(\dfrac{f}{g}\right)(x)$ is:
(1) $[0,1)$ (2) $[0,2)$ (3) $[0,+\infty)$ (4) $[1,+\infty)$

1-7. Which one of the following functions is one-to-one?
(1) $f(x) = x + \sqrt{x}$ (2) $g(x) = x - \sqrt{x}$ (3) $h(x) = 2x + \dfrac{1}{x}$ (4) $p(x) = \dfrac{x}{x^2+1}$

1-8. What is the general solution of the trigonometric equation $\sin 2x \sin 4x + \sin^2 x = 1$?
(1) $k\pi + \dfrac{\pi}{6}$ (2) $(2k+1)\dfrac{\pi}{6}$ (3) $k\pi - \dfrac{\pi}{6}$ (4) $\dfrac{k\pi}{6}$

1-9. What is $\cos^{-1}\!\left(\dfrac{1}{2}\cot\dfrac{11\pi}{3}\right)$?
(1) $-\dfrac{\pi}{3}$ (2) $-\dfrac{\pi}{6}$ (3) $\dfrac{\pi}{3}$ (4) $\dfrac{5\pi}{6}$
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The area enclosed by the curves $y = \sin x + \cos x$ and $y = | \cos x - \sin x |$ over the interval $\left[ 0 , \frac { \pi } { 2 } \right]$ is
(A) $4 ( \sqrt { 2 } - 1 )$
(B) $2 \sqrt { 2 } ( \sqrt { 2 } - 1 )$
(C) $2 ( \sqrt { 2 } + 1 )$
(D) $2 \sqrt { 2 } ( \sqrt { 2 } + 1 )$

Let the functions $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be defined by
$$f ( x ) = e ^ { x - 1 } - e ^ { - | x - 1 | } \quad \text { and } \quad g ( x ) = \frac { 1 } { 2 } \left( e ^ { x - 1 } + e ^ { 1 - x } \right)$$
Then the area of the region in the first quadrant bounded by the curves $y = f ( x ) , y = g ( x )$ and $x = 0$ is
(A) $( 2 - \sqrt { 3 } ) + \frac { 1 } { 2 } \left( e - e ^ { - 1 } \right)$
(B) $( 2 + \sqrt { 3 } ) + \frac { 1 } { 2 } \left( e - e ^ { - 1 } \right)$
(C) $( 2 - \sqrt { 3 } ) + \frac { 1 } { 2 } \left( e + e ^ { - 1 } \right)$
(D) $( 2 + \sqrt { 3 } ) + \frac { 1 } { 2 } \left( e + e ^ { - 1 } \right)$

Consider the functions $f , g : \mathbb { R } \rightarrow \mathbb { R }$ defined by
$$f ( x ) = x ^ { 2 } + \frac { 5 } { 12 } \quad \text { and } \quad g ( x ) = \begin{cases} 2 \left( 1 - \frac { 4 | x | } { 3 } \right) , & | x | \leq \frac { 3 } { 4 } \\ 0 , & | x | > \frac { 3 } { 4 } \end{cases}$$
If $\alpha$ is the area of the region
$$\left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : | x | \leq \frac { 3 } { 4 } , 0 \leq y \leq \min \{ f ( x ) , g ( x ) \} \right\}$$
then the value of $9 \alpha$ is $\_\_\_\_$ .

The area enclosed between the curves $y ^ { 2 } = x$ and $y = | x |$ is
(1) $2 / 3$
(2) 1
(3) $1 / 6$
(4) $1 / 3$

The area (in sq. units) of the region $A = \{(x,y) : (x-1)[x] \leq y \leq 2\sqrt{x},\, 0 \leq x \leq 2\}$, where $[t]$ denotes the greatest integer function, is:
(1) $\frac{8}{3}\sqrt{2} - \frac{1}{2}$
(2) $\frac{4}{3}\sqrt{2} + 1$
(3) $\frac{8}{3}\sqrt{2} - 1$
(4) $\frac{4}{3}\sqrt{2} - \frac{1}{2}$

The area bounded by the curves $y = | x ^ { 2 } - 1 |$ and $y = 1$ is
(1) $\frac { 2 } { 3 } ( \sqrt { 2 } + 1 )$
(2) $\frac { 4 } { 3 } ( \sqrt { 2 } - 1 )$
(3) $2 ( \sqrt { 2 } - 1 )$
(4) $\frac { 8 } { 3 } ( \sqrt { 2 } - 1 )$

The area of the region $\{ ( x , y ) : x ^ { 2 } \leq y \leq | x ^ { 2 } - 4 | , y \geq 1 \}$ is
(1) $\frac { 4 } { 3 } ( 4 \sqrt { 2 } - 1 )$
(2) $\frac { 4 } { 3 } ( 4 \sqrt { 2 } + 1 )$
(3) $\frac { 3 } { 4 } ( 4 \sqrt { 2 } + 1 )$
(4) $\frac { 3 } { 4 } ( 4 \sqrt { 2 } - 1 )$

The area of the region $\{(x, y): x^2 \leq y \leq |x^2 - 4|, y \geq 1\}$ is
(1) $\frac{4(\sqrt{5}-1)}{3} + 4$
(2) $\frac{4(\sqrt{5}-1)}{3} + 2$
(3) $\frac{2(\sqrt{5}-1)}{3} + 4$
(4) $\frac{2(\sqrt{5}-1)}{3} + 2$

The area enclosed between the curves $y = x | x |$ and $y = x - | x |$ is :
(1) $\frac { 4 } { 3 }$
(2) 1
(3) $\frac { 2 } { 3 }$
(4) $\frac { 8 } { 3 }$

The area of the region $\left\{(x, y) : x^2 + 4x + 2 \leq y \leq |x+2|\right\}$ is equal to
(1) 7
(2) 5
(3) $24/5$
(4) $20/3$

The area (in sq. units) of the region $\left\{ ( x , y ) : 0 \leq y \leq 2 | x | + 1,0 \leq y \leq x ^ { 2 } + 1 , | x | \leq 3 \right\}$ is
(1) $\frac { 80 } { 3 }$
(2) $\frac { 64 } { 3 }$
(3) $\frac { 32 } { 3 }$
(4) $\frac { 17 } { 3 }$